Geometry (B)

Inscribed Angles

Learning Objective

• Find the measure of inscribed angles and the arcs they intercept

Inscribed Angle, Intercepted Arc

An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle. An inscribed angle is said to intercept an arc of the circle. We will prove shortly that the measure of an inscribed angle is half of the measure of the arc it intercepts.

Notice that the vertex of the inscribed angle can be anywhere on the circumference of the circle--it does not need to be diametrically opposite the intercepted arc.

Measure of Inscribed Angle

The measure of a central angle is twice the measure of the inscribed angle that intercepts the same arc.

Proof.

and both intercept is a central angle and angle is an inscribed angle.

We draw the diameter of the circle through points and , and let and

We see that is isosceles because and are radii of the circle and are therefore congruent.

From this we can conclude that

Similarly, we can conclude that

We use the property that the sum of angles inside a triangle equals to find that:

and .

Then,

and

Therefore

Inscribed Angle Corollaries a-d

The theorem above has several corollaries, which will be left to the student to prove.

a. Inscribed angles intercepting the same arc are congruent

b. Opposite angles of an inscribed quadrilateral are supplementary

c. An angle inscribed in a semicircle is a right angle

d. An inscribed right angle intercepts a semicircle

Here are some examples the make use of the theorems presented in this section.

Example 1

Find the angle marked in the circle.

The is twice the measure of the angle at the circumference because it is a central angle.

Therefore,

This means that

Example 2

Find the angles marked in the circle.

So,

Example 3

Find the angles marked and in the circle.

First we use to find the measure of angle .

Therefore, .

because they are inscribed angles and intercept the same arc .

In .

Lesson Summary

In this section we learned about inscribed angles. We found that an inscribed angle is half the measure of the arc it intercepts. We also learned some corollaries related to inscribed angles and found that if two inscribed angles intercept the same arc, they are congruent.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

1. In and . Find the measure of each angle:

   

   1. 
   2. 
   3. 
   4. 
   5. 
   6. 
2. Quadrilateral is inscribed in such that .

   

   Find the measure of each of the following angles:

   1. 
   2. 
   3. 
   4. 
3. In the following figure, and .

   

   Find the following measures:

   1. 
   2. 
   3. 
   4. 
4. Prove the inscribed angle theorem corollary a.
5. Prove the inscribed angle theorem corollary b.
6. Prove the inscribed angle theorem corollary c.
7. Prove the inscribed angle theorem corollary d.
8. Find the measure of angle .
   1. 
   2. 
   3. 
   4. 
9. Find the measure of the angles and .
   1. 
   2. 
10. Suppose that is a diameter of a circle centered at , and is any other point on the circle. Draw the line through that is parallel to , and let be the point where it meets . Prove that is the midpoint of .

Review Answers

1. 1. 
   2. 
   3. 
   4. 
   5. 
   6. 
2. 1. 
   2. 
   3. 
   4. 
3. 1. 
   2. 
   3. 
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4. Proof
5. Proof
6. Proof
7. Proof
8. 1. 
   2. 
   3. 
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9. 1. 
   2. 
10. Hint: , so .